Program evaluation with high-dimensional data

In this paper, we consider estimation of general modern moment-condition problems in econometrics in a data-rich environment where there may be many more control variables available than there are observations. The framework we consider allows for a continuum of target parameters and for Lasso-type or Post-Lasso type methods to be used as estimators of a continuum of high-dimensional nuisance functions. As an important leading example of this environment, we first provide detailed results on estimation and inference for relevant treatment eff ects, such as local average and quantile treatment eff ects. The setting we work in is designed expressly to handle many control variables, endogenous receipt of treatment, heterogeneous treatment eff ects, and possibly function-valued outcomes. To make informative inference possible, we assume that key reduced form predictive relationships are approximately sparse. That is, we require that the relationship between the control variables and the outcome, treatment status, and instrument status can be captured up to a small approximation error by a small number of the control variables whose identities are unknown to the researcher. This condition permits estimation and inference to proceed after datadriven selection of control variables. We provide conditions under which post selection inference is uniformly valid across a wide-range of models and show that a key condition underlying the uniform validity of post-selection inference allowing for imperfect model selection is the use of orthogonal moment conditions. We illustrate the use of the proposed methods with an application to estimating the e ffect of 401(k) participation on accumulated assets.